Question

How do you interpret the first derivative?

Answer 1

First derivative of a function

The first derivative of a function let us know much about the function.

If $f\text{'}\left(x\right)$ is positive, then $f\left(x\right)$must be increasing.

If $f\text{'}\left(x\right)$ is negative, then $f\left(x\right)$ must be decreasing.

If $f\text{'}\left(x\right)$ is zero, then $f\left(x\right)$ must be at a relative maximum or relative minimum.

The first derivative of the function $y=f\left(x\right)$ is $f\text{'}\left(x\right)=\frac{dy}{dx}$.

For example, Let $f\left(x\right)=x^2+3$. The first derivative of $f\left(x\right)$ is

$f\text{'}\left(x\right)=2x$. At point $x=3$,

$\begin{array}{rcl}f\text{'}\left(3\right)& =& 2\times 3\\ & =& 6\end{array}$

Hence, the slope of the function $f\left(x\right)$ at point $x=3$ is $f\text{'}\left(3\right)=6$.